Poisson random variable

The Poisson discrete probability function with parameter $\lambda >0$ is given by

$$f_X(x)=\frac{e^{-\lambda }{\lambda }^x}{x!},x\in \mathbb{N}.$$

A random variable $X$ with such a density has expectation, variance, moment generating function and characteristic function given by $E[X]=\lambda$, $Var[X]=\lambda$, $M_X(t)=e^{\lambda (e^t-1)}$, and ${\varphi }_X(t)=e^{\lambda (e^{it}-1)}$, respectively.